Arithmetic or Geometric? a) If a_{1}, a_{2}, a_{3}, \cdots is an

a) To show: the sequence ${\displaystyle a_1+2,a_2+2,a_3+2,\cdots }$ is arithmetic.
Given:
The sequence ${\displaystyle a_1,a_2,a_3,\cdots }$ is an arithmetic sequence.
Approach:
In arithmetic sequence, difference between the terms remains constant throughout.
Calculation:
The sequence ${\displaystyle a_1,a_2,a_3,\cdots }$ is an arithmetic sequence.
${\displaystyle a_2-a_1=a_3-a_2=\cdots \cdots \left(1\right)}$
${\displaystyle (a_2+2)-(a_1+2)=(a_2-a_1)}$
${\displaystyle (a_3+2)-(a_2+2)=(a_3-a_2)}$
From equation (1)
${\displaystyle (a_3+2)-(a_2+2)=(a_2-a_1)}$
As the differences between the terms are same then it is an arithmetic sequence.
Conclusion: hence, the sequence ${\displaystyle a_1+2,a_2+2,a_3+2,\cdots }$ is arithmetic.
b) To show: ${\displaystyle 5a_1,5a_2,5a_3,\cdots }$ is a geometric sequence.
Given: The sequence ${\displaystyle a_1,a_2,a_3,\cdots }$ is geometric sequence.
Approach: In a geometric sequence each term is found by multiplying the previous term by a constant.
Calculation:
${\displaystyle \frac{a_2}{a_1}=\frac{a_3}{a_2}=\cdots \cdots \left(1\right)}$
Multiplying equation (1) by 5
${\displaystyle \frac{5a_2}{5a_1}=\frac{5a_3}{5a_2}=\cdots }$
Then it is a geometric sequence.
Conclusion: hence, the sequence ${\displaystyle 5a_1,5a_2,5a_3,\cdots }$ is a geometric sequence.